Chapter 9 Searching Sorting and Algorithm Analysis Starting

Chapter 9: Searching, Sorting, and Algorithm Analysis Starting Out with C++ Early Objects Ninth Edition by Tony Gaddis, Judy Walters, and Godfrey Muganda Copyright © 2017, 2014 Pearson Education, Inc.

Topics 9. 1 9. 2 9. 3 9. 4 9. 5 9. 6 Introduction to Search Algorithms Searching an Array of Objects Introduction to Sorting Algorithms Sorting an Array of Objects Sorting and Searching Vectors Introduction to Analysis of Algorithms Copyright © 2017, 2014 Pearson Education, Inc. 9 -2

9. 1 Introduction to Search Algorithms • Search: to locate a specific item in a list (array, vector, etc. ) of information • Two algorithms (methods) considered here: – Linear search (also called Sequential Search) – Binary search Copyright © 2017, 2014 Pearson Education, Inc. 9 -3

Linear Search Algorithm Set found to false Set position to – 1 Set index to 0 While index < number of elts and found is false If list [index] is equal to search value found = true position = index End If Add 1 to index End While Return position Copyright © 2017, 2014 Pearson Education, Inc. 9 -4

Linear Search Example • Array numlist contains 17 23 5 11 2 29 3 • Searching for the value 11, linear search examines 17, 23, 5, and 11 • Searching for the value 7, linear search examines 17, 23, 5, 11, 2, 29, and 3 Copyright © 2017, 2014 Pearson Education, Inc. 9 -5

Linear Search Tradeoffs • Benefits – Easy algorithm to understand to implement – Elements in array can be in any order • Disadvantage – Inefficient (slow): for array of N elements, it examines N/2 elements on average for a value that is found in the array, N elements for a value that is not in the array Copyright © 2017, 2014 Pearson Education, Inc. 9 -6

Binary Search Algorithm Binary search requires that the array is in order. 1. Examine the value of the middle element 2. If the middle element has the desired value, done. Otherwise, determine which half of the array may have the desired value. Continue working with this portion of the array. 3. Repeat steps 1 and 2 until either the element with the desired value is found or there are no more elements to examine. Copyright © 2017, 2014 Pearson Education, Inc. 9 -7

Binary Search Example • Array numlist 2 contains 2 3 5 11 17 23 29 • Searching for the value 11, binary search examines 11 and stops • Searching for the value 7, binary search examines 11, 3, 5, then stops Copyright © 2017, 2014 Pearson Education, Inc. 9 -8

Binary Search Tradeoffs • Benefit – It is much more efficient than linear search. For an array of N elements, the number of comparisons is the smallest integer x such that. When N = 20, 000, x is 15. • However, ‒ It requires that the array elements be in order Copyright © 2017, 2014 Pearson Education, Inc. 9 -9

9. 2 Searching an Array of Objects • Search algorithms are not limited to arrays of integers • When searching an array of objects or structures, the value being searched for is a member of an object or structure, not the entire object or structure • Member in object/structure: key field • Value used in search: search key Copyright © 2017, 2014 Pearson Education, Inc. 9 -10

9. 3 Introduction to Sorting Algorithms • Sort: arrange values into an order – Alphabetical – Ascending (smallest to largest) numeric – Descending (largest to smallest) numeric • Two algorithms considered here – Bubble sort – Selection sort Copyright © 2017, 2014 Pearson Education, Inc. 9 -11

Bubble Sort Algorithm 1. Compare 1 st two elements and exchange them if they are out of order. 2. Move down one element and compare 2 nd and 3 rd elements. Exchange if necessary. Continue until the end of the array. 3. Pass through the array again, repeating the process and exchanging as necessary. 4. Repeat until a pass is made with no exchanges. Copyright © 2017, 2014 Pearson Education, Inc. 9 -12

Bubble Sort Example Array numlist 3 contains 17 23 5 11 Compare values 17 and 23. They are in order, so no exchange is needed. Compare values 23 and 5. Exchange them. 17 5 23 11 Compare values 23 and 11. Exchange them. 17 5 11 23 This is the end of the first pass. Copyright © 2017, 2014 Pearson Education, Inc. 9 -13

Bubble Sort Example, Continued The second pass starts with the array from pass one 17 5 11 23 Compare values 17 and 5. Exchange them. 5 17 11 23 Compare 17 and 11. Exchange them. 5 11 17 23 Compare 17 and 23. No exchange is needed. This is the end of the second pass. Copyright © 2017, 2014 Pearson Education, Inc. 9 -14

Bubble Sort Example, Continued The third pass starts with the array from pass two 5 11 17 23 Compare values 5 and 11. No exchange is needed. Compare values 11 and 17. No exchange is needed. Compare values 17 and 23. No exchange is needed. Since no exchanges were made, the array is in order. Copyright © 2017, 2014 Pearson Education, Inc. 9 -15

Bubble Sort Tradeoffs • Benefit – It is easy to understand to implement • Disadvantage – It is inefficient due to the number of exchanges. This makes it slow for large arrays Copyright © 2017, 2014 Pearson Education, Inc. 9 -16

Selection Sort Algorithm 1. Locate the smallest element in the array and exchange it with the element in position 0. 2. Locate the next smallest element in the array and exchange it with element in position 1. 3. Continue until all of the elements are in order. Copyright © 2017, 2014 Pearson Education, Inc. 9 -17

Selection Sort Example Array numlist contains 11 2 29 3 The smallest element is 2. Exchange 2 with the element at subscript 0. Now in order 2 11 Copyright © 2017, 2014 Pearson Education, Inc. 29 3 9 -18

Selection Sort Example, Continued 2 11 29 3 The next smallest element is 3. Exchange 3 with the element at subscript 1. 2 3 29 11 The next smallest element is 11. Exchange 11 with the element at subscript 2. 2 Copyright © 2017, 2014 Pearson Education, Inc. 3 11 29 9 -19

Sorting Considerations • Selection Sort is considered more efficient than Bubble Sort, due to the fewer number of exchanges that occur. • Another efficient sorting algorithm will be seen in Chapter 14. Copyright © 2017, 2014 Pearson Education, Inc. 9 -20

9. 4 Sorting an Array of Objects • As with searching, arrays to be sorted can contain objects or structures • The key field determines how the structures or objects will be ordered • When exchanging the contents of array elements, entire structures or objects must be exchanged, not just the key fields in the structures or objects Copyright © 2017, 2014 Pearson Education, Inc. 9 -21

9. 5 Sorting and Searching Vectors • Sorting and searching algorithms can be applied to vectors as well as to arrays • You need slight modifications to functions to use vector arguments – vector <type> & used in prototype – There is no need to indicate the vector size, as functions can use the vector’s size member function Copyright © 2017, 2014 Pearson Education, Inc. 9 -22

9. 6 Introduction to Analysis of Algorithms • Given two algorithms to solve a problem, what makes one better than the other? • Efficiency of an algorithm is measured by – space (computer memory used) – time (how long to execute the algorithm) • Analysis of algorithms is a more effective way to find efficiency than by using empirical data Copyright © 2017, 2014 Pearson Education, Inc. 9 -23

Analysis of Algorithms: Terminology • Computational Problem: a problem solved by an algorithm • Instance of the Problem: a specific problem that is solved by the algorithm • Size of an Instance: the amount of memory needed to hold the data for the specific problem Copyright © 2017, 2014 Pearson Education, Inc. 9 -24

Analysis of Algorithms: More Terminology • Basic step: an operation in the algorithm that executes in a constant amount of time • Examples of basic steps: – exchange the contents of two variables – compare two values • Non-example of a basic step: – find the largest element in an array Copyright © 2017, 2014 Pearson Education, Inc. 9 -25

Analysis of Algorithms: Terminology • Complexity of an algorithm: the number of basic steps required to execute the algorithm for an input of size N (N = number of input values) • Worst-case complexity of an algorithm: the number of basic steps for input of size N that requires the most work • Average case complexity function: the complexity for typical, average inputs of size N Copyright © 2017, 2014 Pearson Education, Inc. 9 -26

Complexity Example Find the largest value in array A of size n Analysis: 1. biggest = A[0] Lines 1 and 2 execute once. 2. indx = 0 3. while (indx < n) do The test in line 3 executes n times. 4. if (A[n] > biggest) The test in line 4 executes n times. 5. then 6. 7. biggest = A[n] end if 8. end while Copyright © 2017, 2014 Pearson Education, Inc. The assignment in line 6 executes at most n times. Due to lines 3 and 4, the algorithm requires execution time proportional to n. 9 -27

Comparison of Algorithmic Complexity Given algorithms F and G with complexity functions f(n) and g(n) for input of size n • If the ratio approaches a constant value as n gets large, F and G have equivalent efficiency • If the ratio gets larger as n gets large, algorithm G is more efficient than algorithm F • If the ratio approaches 0 as n gets large, algorithm F is more efficient than algorithm G Copyright © 2017, 2014 Pearson Education, Inc. 9 -28

"Big O" Notation • Function f(n) is O(g(n)) (“f is big O of g") for some mathematical function g(n) if the ratio approaches a positive constant as n gets large • O(g(n)) defines a complexity class for the function f(n) and for the algorithm F • Increasing complexity classes means faster rate of growth and less efficient algorithms Copyright © 2017, 2014 Pearson Education, Inc. 9 -29

Chapter 9: Searching, Sorting, and Algorithm Analysis Starting Out with C++ Early Objects Ninth Edition by Tony Gaddis, Judy Walters, and Godfrey Muganda Copyright © 2017, 2014 Pearson Education, Inc.